Nonlinear Dynamics and Quantum Chaos - An Introduction

Foreword to the First Edition
Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
1.1 Fundamental Terminology
1.2 Complexity
1.3 Classical Versus Quantum Dynamics
Problems
2 Dynamical Systems
2.1 Evolution Law
2.2 One-Dimensional Maps
2.2.1 The Logistic Map
2.2.2 The Dyadic Map
2.2.3 Deterministic Random Number Generators
Problems
3 Nonlinear Hamiltonian Systems
3.1 Integrable Examples
3.2 Hamiltonian Formalism
3.3 Important Techniques in the Hamiltonian Formalism
3.3.1 Conserved Quantity H
3.3.2 Canonical Transformations
3.3.3 Liouville's Theorem
3.3.4 Hamilton-Jacobi Theory and Action-Angle Variables
3.4 Integrable Systems
3.4.1 Examples
3.4.2 Poisson Brackets and Constants of Motion
3.5 Non-Integrable Systems
3.6 Perturbation of Low-Dimensional Systems
3.6.1 Adiabatic Invariants
3.6.2 Principle of Averaging
3.7 Canonical Perturbation Theory
3.7.1 Statement of the Problem
3.7.2 One-Dimensional Case
3.7.3 Problem of Small Divisors
3.7.4 KAM Theorem
3.7.5 Example: Two Degrees of Freedom
3.7.6 Secular Perturbation Theory
3.8 Transition to Chaos in Hamiltonian Systems
3.8.1 Surface of Sections
3.8.2 Poincaré-Cartan Theorem
3.8.3 Area Preserving Maps
3.8.4 Fixed Points
3.8.5 Poincaré–Birkhoff Theorem
3.8.6 Dynamics near Unstable Fixed Points
3.8.7 Mixed Regular-Chaotic Phase Space
3.9 Criteria for Local and Global Chaos
3.9.1 Ergodocity and Mixing
3.9.2 The Maximal Lyapunov Exponent
3.9.3 Numerical Computation of the Maximal Lyapunov Exponent
3.9.4 The Lyapunov Spectrum
3.9.5 Kolmogorov-Sinai Entropy
3.9.6 Resonance Overlap Criterion
4 Dissipative Systems
4.1 Introduction
4.2 Fixed Points
4.2.1 Fixed Point Scenarios in Two-Dimensional Systems
4.3 Damped One-Dimensional Oscillators
4.3.1 Harmonic Oscillator
4.3.2 Nonlinear Oscillators
4.3.3 Nonlinear Damping
4.4 Poincaré–Bendixson Theorem
4.5 Damped Forced Oscillators
4.5.1 Driven One-Dimensional Harmonic Oscillator
4.5.2 Duffing Oscillator
4.6 Lorenz Model
4.7 Fractals
4.7.1 Simple Examples
4.7.2 Box-Counting Dimension
4.7.3 Examples from Nature
4.8 Bifurcation Scenarios
4.8.1 Examples of Pitchfork Bifurcations
4.8.2 Tangent Bifurcations
4.8.3 Transcritical Bifurcations
4.8.4 Higher-Order Bifurcations
4.8.5 Hopf Bifurcations
4.9 Two Routes to Chaos
4.9.1 Landau's Transition to Chaos
4.9.2 Ruelle–Takens–Newhouse Route to Chaos
4.10 Intermittency
4.11 Coupled Oscillators
4.11.1 Circle Map
4.11.2 Arnold Tongues and Farey Trees
4.11.3 Synchronization
4.11.4 Kuramoto Model
4.12 Increasing Complexity
Problems
5 Aspects of Quantum Chaos
5.1 Introductory Remarks on Quantum Mechanics
5.2 Semiclassical Quantization of Integrable Systems
5.2.1 Bohr–Sommerfeld Quantization
5.2.2 Wentzel–Kramer–Brillouin–Jeffreys Approximation
5.2.3 Einstein–Brillouin–Keller Quantization
5.2.4 Semiclassical Wave Function for Higher-Dimensional Integrable Systems
5.3 Semiclassical Description of Non-Integrable Systems
5.3.1 Green Functions
5.3.2 Feynman Path Integral
5.3.3 Method of Stationary Phase
5.3.4 Van Vleck Propagator
5.3.5 Semiclassical Green Function
5.3.6 Gutzwiller's Trace Formula
5.3.7 Applications of Semiclassical Theory
5.4 Wave Functions in Phase Space
5.4.1 Phase-Space Densities
5.4.2 Weyl Transform and Wigner Function
5.4.3 Localization Around Classical Phase-Space Structures
5.5 Anderson and Dynamical Localization
5.5.1 Anderson Localization
5.5.2 Dynamical Localization in Periodically Driven Quantum Systems
5.5.3 Experiments
5.6 Universal Level Statistics
5.6.1 Level Repulsion: Avoided Crossings
5.6.2 Level Statistics
5.6.3 Symmetries and Constants of Motion
5.6.4 Density of States and Unfolding of Spectra
5.6.5 Nearest-Neighbour Statistics for Integrable Systems
5.6.6 Nearest-Neighbour Statistics for Chaotic Systems
5.6.7 Gaussian Ensembles of Random Matrices
5.6.8 More Sophisticated Methods
5.7 Concluding Remarks
Index
Index